Abstract: Let be a unital Banach algebra. We give a characterization of the left Banach -modules for which there exists a commutative unital -algebra , a linear isometry , and a contractive unital homomorphism such that for any . We then deduce a ``commutative" version of the Christensen-Effros-Sinclair characterization of operator bimodules. In the last section of the paper, we prove a -version of the latter characterization, which generalizes some previous work of Effros and Ruan.